3.50 \(\int \frac{x}{\sqrt{a^4-x^4}} \, dx\)

Optimal. Leaf size=22 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x^2}{\sqrt{a^4-x^4}}\right ) \]

[Out]

ArcTan[x^2/Sqrt[a^4 - x^4]]/2

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Rubi [A]  time = 0.0238282, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{2} \tan ^{-1}\left (\frac{x^2}{\sqrt{a^4-x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a^4 - x^4],x]

[Out]

ArcTan[x^2/Sqrt[a^4 - x^4]]/2

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Rubi in Sympy [A]  time = 1.60058, size = 15, normalized size = 0.68 \[ \frac{\operatorname{atan}{\left (\frac{x^{2}}{\sqrt{a^{4} - x^{4}}} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a**4-x**4)**(1/2),x)

[Out]

atan(x**2/sqrt(a**4 - x**4))/2

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Mathematica [A]  time = 0.00739993, size = 22, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (\frac{x^2}{\sqrt{a^4-x^4}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a^4 - x^4],x]

[Out]

ArcTan[x^2/Sqrt[a^4 - x^4]]/2

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Maple [A]  time = 0.016, size = 19, normalized size = 0.9 \[{\frac{1}{2}\arctan \left ({{x}^{2}{\frac{1}{\sqrt{{a}^{4}-{x}^{4}}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a^4-x^4)^(1/2),x)

[Out]

1/2*arctan(x^2/(a^4-x^4)^(1/2))

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Maxima [A]  time = 1.48702, size = 24, normalized size = 1.09 \[ -\frac{1}{2} \, \arctan \left (\frac{\sqrt{a^{4} - x^{4}}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(a^4 - x^4),x, algorithm="maxima")

[Out]

-1/2*arctan(sqrt(a^4 - x^4)/x^2)

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Fricas [A]  time = 0.223773, size = 34, normalized size = 1.55 \[ -\arctan \left (-\frac{a^{2} - \sqrt{a^{4} - x^{4}}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(a^4 - x^4),x, algorithm="fricas")

[Out]

-arctan(-(a^2 - sqrt(a^4 - x^4))/x^2)

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Sympy [A]  time = 1.74402, size = 29, normalized size = 1.32 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{x^{2}}{a^{2}} \right )}}{2} & \text{for}\: \left |{\frac{x^{4}}{a^{4}}}\right | > 1 \\\frac{\operatorname{asin}{\left (\frac{x^{2}}{a^{2}} \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a**4-x**4)**(1/2),x)

[Out]

Piecewise((-I*acosh(x**2/a**2)/2, Abs(x**4/a**4) > 1), (asin(x**2/a**2)/2, True)
)

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GIAC/XCAS [A]  time = 0.216464, size = 14, normalized size = 0.64 \[ \frac{1}{2} \, \arcsin \left (\frac{x^{2}}{a^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(a^4 - x^4),x, algorithm="giac")

[Out]

1/2*arcsin(x^2/a^2)